# Beam pattern for a linear array comprised of rectangular transducers

I am looking for the beam pattern of a linear array with uniform spacing, $d$, between $n$ elements. I know for an array comprised of circular pistons it can be derived from

$$\frac{1}{n}\frac{\sin[n\pi\frac{d}{\lambda}(\sin\theta-\sin\theta_s)]}{\sin[\pi\frac{d}{\lambda}(\sin\theta-\sin\theta_s)]}$$

where $\lambda$ is the wavelength. And it is a narrowband system. I appreciate if you can provide a reference book and describe how I can derive the beam pattern for the particular case of rectangular elements.

• Though antenna design, communication engineering and signal processing aren't really disjunct, this is really a question that should be asked on electronics.stackexchange.com, as it's not concerned with processing the signals involved. – Marcus Müller Apr 5 '17 at 16:48
• Well, I thought as most of the beamforming is applied to the signals in the phase space, someone might know about time space as well. But, thanks for the comment. I submit it to electronics. – Mahdi Razaz Apr 5 '17 at 19:19
• You have to look for the term "Array Factor" for your specific antenna type i.e Rectangular. Then you have to multiply this factor with the beam pattern of the Uniform Linear Array (ULA) with isotropic radiators as its element. This will in terms give you the complete beam pattern of your array. You can also look into first chapters of book "Fundamental of Radar Signal Processing" by Mark A. Richards which states the same. – Zeeshan Apr 6 '17 at 9:37
• It's not working. I already did that. The above equation is the AF for a ULA which can steer the main lobe theta_s degree. multiplying it by a rectangular piston AF destroys the whole beam pattern and doesn't have any steering ability. At theta_s=0 it works well but for any steering angle, funny things happen. – Mahdi Razaz Apr 6 '17 at 13:00
• The normal case is to multiply the Array Factor (ULA) with the typical element pattern to achieve the composite radiation pattern. This is as stated in the book referenced above. I dont have a clue why it is not working in your case, hope that some one with more insight can answer :) – Zeeshan Apr 7 '17 at 13:16